CC BY
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Вычислительные технологии
Том 6, № 2, 2001
STRUCTURAL STABILITY IN GENERALIZED SEMI-INFINITE OPTIMIZATION*
G.-W. Weber Department of Mathematics Darmstadt University of Technology, Germany e-mail: weber@mathematik.tu-darmstadt.de
Обсуждаются свойства многообразий и вопросы непрерывости допустимых ограничений М[Н, д, и, ь], а также соответствующее поведение (/, Н, д, и, у) при слабых возмущениях. Формулируются теоремы о многообразиях, непрерывности, универсальности, устойчивости и структурной устойчивости. Кратко описываются возможные расширения на случаи неограниченности и недифференцируемости, указываются такие структурные границы, при которых полученные результаты могут трактоваться в терминах задач оптимального управления для обыкновенных дифференциальных уравнений.
1. Introduction
Under suitable assumptions, the following fields of problems from science, engineering and control lead to generalized semi-infinite (GSZ) optimization: o optimizing the layout of a special assembly line, o maneuverability of a robot, o time minimal heating or cooling of a ball of some homogeneous material, o approximation of a thermo-couple characteristic in chemical engineering, o structure and stability in optimal control of ordinary differential equations. For motivations and references see, e.g., [59, 60]. In future, GSI applications may also be expected in optimal experimental design ([9]). The GSI problems under consideration have the form
Minimize f(x) on Mgsi[h,g], where
Vgsi(f,h,g,u,v) { Mgsi[h,g] := { x G Rn | h(x) = 0 (i G I),
g(x,y) > 0 (y G Y(x)) }.
The semi-infinite character comes from the perhaps infinite number of elements of Y(= Y(x)) [10, 45], while the generalized character comes from the x-dependence of Y(•). We suppose these index sets to be finitely constrained (F):
Y(x) = mf[u(x, •), v(x, •)] := { y G IRq | uk(x,y) = 0 (k G K), v^(x,y) > 0 (t G L)}(x G IRn).
Let h = (hi)iGl, u = (uk)keK, v = (v^eL, where hi : IRn ^ IR, i G I := {1,... ,m}, Uk : Rn x Rq ^ R, k G K := {1,... , r}, v£ : Rn x Rq ^ R, t G L := {1,...,s} (m < n; r < q).
G.-W. Weber, 2001
We assume that f : Rn ^ R, g : Rn x Rq ^ R, hi (i E I), uk (k E K), ve (£ E L) are once continuously differentiable (CBy Df (x), DTf (x) we denote the row- (column) vector
d d of the first-order partial derivatives ^— f (x), and Dxg(x,y), Dyg(x,y) consist of ^— g(x,y)
d K K and —— g(x,y). Let a given set U0 C Rn, MgSi[h,g] HU0 = 0, be bounded and open. oya
Assumption Ayo: Uxeyo Y(x) is bounded (hence, by continuity, compact). In generalized semi-infinite optimization, the feasible set MgSi[h,g] need not be closed [24]. The following assumption, however, ensures closedness:
Assumption Byo: For all x eU0, the linear independence constraint qualification (LICQ) is fulfilled for Mf[u(x, •),v(x, •)], i.e., linear independence of
DyUk(x,y), k E K, Dyv£(x,y), £ E Lo(x,y)
(considered as a family), where L0(x,y) := { £ E L | ve(x,y) = 0 } consists of active indices.
Using differential topology [17, 20], these assumptions admit local linearization of Y(x) (x E U0) by finitely many C:-diffeomorphisms j : Vj ^ Sj (j E J) in such a way that the image sets Zj of indices are x-independent squares (in a linear subspace). Herewith, PgSi(f,h,g,u,v) becomes locally (in U0) equivalently expressed as an ordinary semi-infinite optimization problem POSi(f,h,g0,u0,v0), where MOSl[h,g0] HU0 = MgSi[h,g] HU0, f being unaffected [57, 59].
On the upper stage of variable x, we shall use a constraint qualification, too. This cq geometrically means the existence of an (at M[h] = h-1({0})) tangential, "inwardly" pointing direction at x:
Definition. We say that the extended Mangasarian-Fromovitz constraint qualification (EMFCQ) is fulfilled at a given x E MgSi[h,g], if the conditions EMF1t2 are satisfied:
EMF 1. Dhi(x), i E I, are linearly independent. EMF2. There exists an "EMF-vector" Z E Rn such that
Dhi(x) Z = 0 for all i E I,
Dxg0(x,z) Z > 0 for all z E Rq,j E J, with (j)-1(z) E Y0(x),
where Y0(x) := { y E Y(x) | g(x,y) = 0 } consists of active indices. EMFCQ is said to be fulfilled for Mgsi[h,g] on U0, if EMFCQ is fulfilled for all x E Mgsi[h,g] H U0.
For further information and versions of EMFCQ see [15, 20, 24, 26, 40], but also [7, 18]. Let a local minimum x of PgSi(f, h, g, u, v) be given and EMFCQ be fulfilled there. Then, we can state the existence of Lagrange multipliers Ai, such that the conditions
Df (x) = XiDhi(x) + »KDxg0K (x,zK),
iel k€{1,...,K}
Vk > 0 (K E {1,...,K})
are satisfied, referring to ordinary semi-infinite (OSI) data [15, 57, 59]. Now, we call x a G-O Kuhn-Tucker point. Here, the points zk E Zj are suitable active indices. Referring to all the given GSI data now, a further evaluation yields the following Kuhn-Tucker conditions with corresponding Lagrange multipliers Ai, /jk, aK}k, PK,e [57, 59]:
KTi. Df(x)
J2\Dhi(x) + ^ ^KDxg(x,yK) -
iel k€{1,...,K}
- S aK,kDuk(X,yK) - Y, PK,£DxVt(X,yK),
k&K
t&L0(x,yK) k£{1.....K}
KT2. > 0 G Lo(x,yK),K G {1,... , k}).
Again, the yK G Y°(X) are active. Now, we call X a G Kuhn-Tucker point. Under general assumptions, the necessary optimality condition KT1; 2 was for the first time proved by Jongen, Ruckmann and Stein [24]. Note, that the linear combination KT1 contains the derivatives of all the defining functions. The foregoing conditions can also be stated as growth (angular) conditions over tangent cones [32, 57, 59]. These growth conditions estimate scalar products against 0; they give rise to deduce first-order sufficient optimality conditions. In fact, let LICQ be satisfied at a given point x as an element of M[h], and M[h] flU° be star-shaped with star point X. Moreover, let the functions g°(-,z) (z G Zj, j G J) be quasi-concave and f be pseudo-convex on M[h] flU°. This means the following implications for all x G M[h] If U° [16, 32]:
g0 (x,z) > gj(x,z) = Df (x) (x - x) > 0
Dxg0(x, z) (x - x) > 0, f (x) > f (x).
Then, X turns out to be a local minimizer of VgSI(f,h,g,u,v) [57, 59]; cf. [29]. Concerning structural frontiers in (F) nonconvex optimization see [28]. Before we introduce the second-order condition of strong stability we state under our basic Assumptions Ay0, By0:
Lemma [59]. Let X G MgSI[h,g] fU0 be given, and EMFCQ be fulfilled at X. Then, X is a G-O Kuhn-Tucker point for VgSI(f,h,g,u,v), if and only if the extended Mangasarian-Fromovitz constraint qualification on MgSI[h, (g, —f + f (X))], called EMFCQ, is violated at X.
Proof: This result results from Farkas' Lemma for infinite systems [15, 53, 59]. I We prepare our introduction of strong stability of a stationary point by assuming that f,h,g,u,v are C2 and putting for any bounded open neighbourhood V C ]Rq of (J Y(x)
and any subset M C ]Rn:
normgsj[(f, h,g,u,v), M] :=
sup
sup max
x&M ^{f}^ {hv\ve I}
|y (x)| +
El & (x)l
+ y^ l d2Y (x
I dxidxn v
j=1
sup max
iE M n6{9}U y6V {uv |v6K}U {vv |vG L}
n
ln(x)| + E lg(x,
i=l
+ E l gj(x,
j=l
n d^
+ E 1 dx.dx, (x)! +
j=l
n q d2v q d2n
+ E E 1 dxidy, (x)| + E 1 oyiay,(x
i=lj=l i=1
j j=1
In cases of F or OSZ optimization, replacing V by J, Y or disregarding u,v, we denote by normF[■, ■], normOSI[■, ■]. Because of continuity properties stated in Section 2, the next condition is well-defined [59].
Definition. Suppose a feasible point xu E MgSi[h,g] HU0 for PgSi(f,h,g,u,v) (of class C2). Now, VoSi(f,h,g0,u0,v0) be locally (in U0) representing VgSi(f,h,g,u,v), and xu be a G-O Kuhn-Tucker point of PgSi(f,h,g,u,v). Then, we say that xu is (G-O) strongly stable, if for some e > 0 with B(xu,e) C U0 and for each e E (0,e] there is some 5 > 0 such that for each C2-function (f ,h,g0) with normOSi[(f — f ,h — h,g0 — g0),B({cu,t)\ < 5 the open ball B(xu,e) contains an ordinary Kuhn-Tucker point xd of VqS1 (f ,h,g0) := VoSi(f ,h,g0,u0,v0), which is unique in B(xu,e). Referring to a G Kuhn-Tucker point xu and to normgSi[(f -f ,h - h,g - g,u - u,v - v),B({cu,t)\, we get the condition of (G) strong stability of xu.
Here, "u" (and "d") indicates (un)disturbed. For our preferred (G-O) strong stability expressed by original GSI data, see [59]. In Section 3, we utilize an algebraical characterization of strong stability in the tradition of Kojima [30] and Ruckmann [48].
2. Stability of the Feasible Set.
The following theorems underline the importance of EMFCQ for concluding that MgSi[h,g,u,v] := MgSi[h,g] is a topological manifold with boundary, it behaves continuous and stable under perturbations of our defining C1-functions. With these perturbations we remain inside of suitable open neighbourhoods of (h, g, u, v) in the sense of the strong or Whitney topology CS1 that takes into account asymptotic effects (for topologies CSk, k E IN U {to}, cf. [17, 20]). We call a given subset M C Rn a Lipschitzian manifold (with boundary) of dimension k, if for each x E M there are open neighbourhoods W1 C Rn of x, W2 C Rn of 0n, and a bijective "chart" < : W1 ^ W2, <(x) = 0n, with Lipschitzian continuity of <-1 such that < carries MHW1 to the relatively open set ({0n-K}x Rk)HW2 or to the set ({0n-K}x {w E R | w > 0}x Rk-1)HW2 with (relative) boundary. So, Lipschitzian manifolds can locally be linearized, however, without preserving "angulars" in the boundary. In F optimization, that preservation is guaranteed by the stronger condition LICQ, using C1-smooth linearizing charts. In this sense, we find qualified versions of the following topological results for Y(x), [19, 59]. _
Manifold Theorem [59]. LetEMFCQ be fulfilled in U0 for Mgsi [h,g]. Then, there is an open neighbourhood W C Rn of U0 such that MgSi [h,g] HW is a Lipschitzian manifold (with boundary) of the dimension n - m. Moreover, then we can represent the (relative) boundary:
(dMgsi [h,g]) HW = { x E W| hi(x) = 0 (i E I), min g(x,y) = 0 }.
y&Y (x)
Proof: Assumption Byo, delivers diffeomorphisms fy, for all x of some open neighbourhood W of U0. These transformations guarantee that the insight from [26] on OSI optimization can be carried over for our GSI problem.
For the properties of upper and lower semi-continuity, continuity (in Hausdorff-metric), genericity (implying density) and transversality (absense of tangentiality), considered for functions or sets next, we refer to [3, 17, 20, 26, 59].
Continuity Theorem [58, 59]. Let EMFCQ be fulfilled in U0 for Mgsi[h, g]. Moreover, let the closure W C Rn of some open set W C U0 be representable as a feasible set from F optimization which fulfills LICQ, and let the intersection of its boundary dW with MgSi[h,g] be transversal. Then, there is an open C^-neighbourhood O C (C1 (Rn,R))mxC 1(IRn+q,R)x (C1 (Rn+q, R))r x(C1 (Rn+q, R))s of (h,g,u,v) such that MW : (h,g,u,v) ^ Mgsi[h,g,u,v] H W, is upper and lower semi-continuous at all (h,g,u,v) E O. If, moreover, W is bounded, then O can be chosen so that O is mapped to Vc(Rn) by MW, and MW is continuous.
Proof: These assertions are consequences of the continuous dependence of the OSI functional data g0,u0,v0 on the GSI data g,u,v and, then, of [26], Theorem 2.2. We apply this theorem on MOSi[h,g0,u0,v0] := MOSi[h,g0]. In the proof of Genericity Theorem below, we investigate the continuous dependence : (h,g,u,v) ^ (h,g0,u0,v0).
In [59], also a global version and a version on (x,u,v) ^ Yu'v(x) are presented for the previous theorem. The following result refers to the straightforward generalization ELICQ of LICQ that is a stronger condition than EMFCQ [26, 53, 59]. (The double usage of F should not lead to any confusion. For a global result see [59].)
Genericity Theorem [59].
(a) Let C™ := (C™(Rn ,R))m xC™(Rn xRq ,R)x(C™(Rn xRq, R))r x(C™(RnxRq ,R))s be endowed with the C™-topology. Furthermore, let its subspace Cj^c of all (h,g,u,v) E C™ with validity of Assumptions Ayo, Byo be endowed with the C™-relative topology. Then, there exists a generic subset E C C™c such that ELICQ is satisfied for each (h,g,u, v) E E.
(b) Let C1 := (C1(Rn,R))m x C1(Rn xRq ,R) x (C1(Rn xRq, R))r x(C1(Rn xRq ,R))s be endowed with the C1 -topology. Furthermore, let its subspace Cj1 of all (h,g,u,v) E C1 with validity of Auo, Buo be endowed with the C\-relative topology. Then, there exists an open and dense subset F C C|oc such that EMFCQ is satisfied for each (h,g,u,v) E F. The set F can just be defined by the fulfillment of EMFCQ.
Outline of Proof: The first insight on the desired subset E of C™-functions follows from the OSI result [26], Theorem 2.4, that applies Multi-Jet Transversality Theorem [17, 20] and additional reflections. For that theorem our u0, v0 are kept fixed, focussing topological interest on (h,g0) (h = h0); here, the part of some constant set Y is taken by the union of the sets Zj (j E J). Without loss of generality, J consists of one single element. Now, we can state that there is a generic set EO of OSI data functions (h,g0), which (by definition of genericity) is the intersection of countably many open and dense subsets EO'v (v E IN).
However, for the tracing back of the OSI genericity (or, below, openess and density) to GSI optimization, we utilize that the problem representation is continuous. In fact, by Implicit Function Theorem in Banach Spaces [20, 37], the inserted local coordinate transformations continuously depend on (h, V, u, vv). Let us regard this continuous dependence (representation) as a function locally mapping (h,V,u,v) E C™ into the space of all C™-functions (h,V0,u0,v0). With respect to h, the mapping is constant. Using we find E as the intersection of the countably many open sets Ev := ^-1(EO,v) (v E IN).
Now, let us consider an element (h,g,u,v) E C™c. After sufficiently small perturbations it still remains in Cj^c. Let also some v E IN be given. In the OSI problem, however, we consider
separate (de-coupled) perturbations gj0 ^ gj (j E J) (before we really turn to one single inequality j). Therefore, the "problem representation" is not surjective. Actually, as for some x EU0 and two (or more) different j 1,j2 E J the sets (j)-1(Z0 (x)), (^x)-1(Zj (x)) might have a nonempty intersection, these perturbations cannot always be traced back to a perturbation g ^ V of the given GSI problem. The following perturbation technique, however, will be helpful to get rid with such a difficulty, and it will finally guide us to the asserted density.
By definition of rfje (j E J) (linearization) the implicitly disturbed sets Zj can be chosen as Zj. Moreover, because of the locally finite covering structure underlying no difficulty arises. In view of that locally "fix" u0,v0 and of the constant property of with respect to h, we delete u0,v0,h in the definition of . So, we get a mapping called First of
all, we add to g one j-independent, arbitrarily CSf-small positive function g in an arbitrarily
small neighbourhood of the compact set (J (^)-1(Z° (x)) f(4>XX)-1(Z° (x)), making
x&Mgsx [h,g]n U°
active indices y inactive there. Then, g* := g + g is a globally defined C^-function. Now, for each v G IV we find a (componentwise) arbitrarily C^-close approximation (hv, gv°, uv°, vv°) G E°'v of (h°,g°,u°,v°), where the approximation gv° coincides with g*° := ^R(g*,u,v) in UjejZj. Here, we may choose the C 1-function (uv°,vv°) := (u°,v°). Hence, that perturbed function gv° can continuously be traced back under ^R-1 to one C^-function gv, i.e., {(gv,u,v)} = ^R-1 ({gv°}). So we are in a position to state, that (h,g,u,v) can arbitrarily well be C^-approximated by (h, g,u,g) := (hv,gv,u,v) G Ev. This means that Ev is dense, too. Altogether, we have shown that E is generic.
Preparation: This (relative) genericity implies (relative) density [20], because of the " C openess" of both LICQ and (y-) boundedness. Now, we use the fact that EMFCQ follows from ELICQ, and the C^-density of C~(IRfc, IR) in C 1(]Rk, IR) (k G IV). Moreover, we take account of our preparation and of the perturbational "C^-openess" of EMFCQ.
We underline "F" or "GSI open" properties: LICQ and EMFCQ remain preserved under sufficiently slight data perturbation.
Next, we refer to the same underlying dimensions n, q in x- or y-space, and numbers r, s of functions uk, v£. Two feasible sets MgSI[h1 ,g1,u1 ,v1 ], MgSI[h2,g2,u2, v2] are called (topologically) equivalent, notation: MgSI[h1,g1,u1 ,v1] ~M MgSI[h2,g2, u2,v2], if there is a homeomorphism (= <fiMgsi) : IRn ^ IRn such that
^M (MgsI [h1 ,gV ,v1 ]) = Mgsi [h2,02,uV].
The given feasible set MgSI[h,g] (= MgSI[h,g,u,v]) is called (topologically) stable, if there is an open C^-neighbourhood O of (h,g,u,v) such that for each (h, g,u, v) G O we have MgSI[h,g,u,v] ~M MgSI[h, g,u,v] (see [12, 26, 53, 59]. Let us make the boundedness (hence, compactness) assumption that MgSI[h,g] lies in U°.
Stability Theorem [58, 59]. The feasible set MgSI[h,g] C U° is topologically stable, if and only if EMFCQ is fulfilled for MgSI[h,g].
Proof: We trace back to the OSI situation again, given by [26], Theorem 2.3, now. As being the case in the proof of Genericity Theorem, technicalities arise. Moreover, in [26] the equality constraint functions h are assumed to be C2. All these difficulties can be governed: In Section 3 we prove Characterization Theorem on the lower level sets of the whole GSI optimization problem; that theorem implies our Stability Theorem. We note that under our overall boundedness assumptions, MgSI[h,g] is a lower level set of PgSI(f, h,g,u,v) for a sufficiently high f-level. Already to point out the essential ideas for the sufficiency part, "proved in a constructive way, and for the necessity part, " =^", proved in an indirect way, we look at Figures 1, 2, respectively. For both parts differential topology and Morse theory are helpful. While for the necessity part some algebraic topology [19, 51] is essential to evaluate unstable situations, for the sufficiency part flows [1] are important. To construct a homeomorphism , we first of all C^transform (in U) the sets M[h],M[h] to some manifold M[h]. Here h is of class C2 (or C^ ) [47, 53]. Now, we may suppose I = 0. Finally, we homeomorphically map the feasible set MgSI[g] onto the feasible set MgSI [g] by steering the boundary dMgSI[g] onto dMgSI[g] along an EMF-vector field.
a
/
/
i Mih]
L J
I
M[h] I
Fig. 1. Proof of sufficiency part, Stability Theorem
Mgsi[h,g,u,v]
\MQSl[h,g,u,v]
Fig. 2. Proof of necessity part, Stability Theorem
3. Structural Stability and its Characterization. 3.1. Structural Stability of the Problem.
Under Assumptions Auo, Buo, we still refer to the bounded set Mgsi[h,g], but additionally take f into consideration. We establish the structure of the entire problem Pgsi(f,h,g,u,v) by all its lower level sets
LTgsi(f, h, g, u, v) := { x E Rn | x E Mgsi[h,g,u,v], f (x) < t } (t E R).
In the tradition of Guddat, Jongen, Riickmann and Weber, we observe this structure under data perturbation and define structural stability. Here, descent has to be preserved, if the level varies. Let us still assume that the defining functions are C2. Then, this global stability can essentially be characterized by EMFCQ of Mgsi[h,g] and by strong stability of all considered stationary points.
Two problems Pgsi(f 1,h1, g1,u1,v1), Pgsi(f2 ,h2, g2 ,u2,v2) (with defining C2-functions) are called structurally equivalent:
Vgsi(f1,h1,g1,u1,v1) -P Pgsi(f2,h2,g2,u2,v2)
if there are continuous functions <fp (= <fipgsi) : R x Rn ^ Rn and ^ (= ^gsi) : IR ^ IR with the three properties Egsi 1t 2t 3 (Fig. 3):
Egsi 1. <p,T : Rn ^ Rn is a homeomorphism, where <p,T(x) := <p (t,x), for every t E IR. Egsi 2. ^ : IR ^ IR is a monotonically increasing homeomorphism.
Fig. 3. Structural equivalence (bird's-eye view below)
Egsi 3. (LTsi (f 1,h1,g1,u1,v1)) = L$x(f2, h2,g2,u2 ,v2) for all t G IR.
Considering the first problem as undisturbed and the second one as slightly disturbed, we arrive at structural stability [11, 23, 27, 53, 59]; cf. also [1, 4, 20, 50]: Vgsi(f, h, g,u,v) (with defining C2-functions) is called structurally stable, if there exists a C 2-neighbourhood O of (f, h,g,u,v) such that for each (f,h, g,u,g) GO
Pgsi(f, h, g, u, v) PgSi(f, h,g,u,g)
3.2. Characterization Theorem.
Under Assumptions AMo and BMo we state:
Characterization Theorem (or Structural Stability Theorem; [59]). Let Mgsi[h,g] CU° hold for problem PgSI(f, h, g,u,v) (with defining C2-functions). Then, PgSI(f, h,g,u,v) is structurally stable, if and only if the three conditions CgSI 1,2,3 are fulfilled:
Cgsi 1. EMFCQ holds for Mgsi[h, g].
CgSI2. All the G-O Kuhn-Tucker points x of PgSI(f, h,g,u,v) are (G-O) strongly stable.
CgSI3. For each two different G-O Kuhn-Tucker points X1 = X2 of VgSI(f, h,g, u,v) the corresponding critical values are different (separate), too: f (x1) = f (x2). In this main result, we could also make a further assumption, excluding certain inequality constraints z from the relative boundary dZj (j G J). Then we could identify the G-O Kuhn-Tucker points by some G Kuhn-Tucker points. However, for validity of Characterization Theorem, such an assumption is not necessary [59].
3.3. Proof of Characterization Theorem.
Preparations. For preparation, let us recall the proof of Genericity Theorem, taking into account the parametrical dependences on the defining data (g , u, v ) (by construction, h may be disregarded). Now, we make again applications of Implicit Function Theorem in Banach spaces, such that, in particular, we state a continuous dependence of (0°^°^°) on (g,u,v ). Consequently, small perturbations on the data of PgSI(f, h,g,u, v) cause slight perturbations on the data of POSI(f, h,g°, u°, v°). The reverse question arises: Can small perturbations of the OSI data be reconstructed under the problem representation from slight perturbations of
the given GSI problem ? We give a conditionally positive answer. However, this answer will be fitting for the perturbational argumentations on Characterization Theorem:
Item 1. For representing OSI problem(s), u°,v° are of special linearly affine form and, under sufficiently small perturbations of the GSI problem, we may treat them as fixed. Hence, besides the perturbations (f,h) ^ (f ,h), for VoS1 (f,h,g°,u°,v°) we are concerned with g° ^ g° only. We therefore introduce the simplifying notation VqS1 (f, h, g°) := VoS1 (f, h, g°,u°,v°).
Item 2. Subsequently, we mainly perform local perturbations for VqS1 (f,h,g°). Hereby, we treat the finitely many functions gjj (j E J) separately in small disjoint open sets
V* (j E J) such that their perturbations gjj ^ gj can be reconstructed by one single C2-function g (given below). Therefore, we would need the perturbationally stable Assumption F *: For all j1,j2 E J, j1 = j2, we have
U
(j)-1(zo (x)) n (j)-'(z0 (*))| =
•Jx ) w;1 1 Wx )
x&Mgsi[h,g]
We are going to exploit the condition from Assumption F* after perturbations. However, if we may suitably choose our perturbed functions g0, then Assumption F* is naturally fulfilled (after perturbation), and we need not make it in the unperturbed situation. Now, under problem representation and joined by u,v, this function g generates gj locally in V* (j E J). Then, for each j E J, small perturbational (global) effects outside of V* (j E J) have no influence. They can be ignored. The announced function is
g (x,y)
g0(x,jy)), if y E (<&)-1(Zj) and (x,jy)) E V*, j E J g(x,y), else.
Item 3. Below we must consider a certain global perturbation of POSl(f, h,g0) to receive C -data or, finally, some (global) "open and dense" property. Therefore, we apply on the one hand the perturbation technique from the proof of Genericity Theorem. On the other hand, whenever it is possible to turn from the GSI problem to an OSI (or F) one, then we are back in the situation of Item 2 in order to perform local perturbations.
For our proof of Characterization Theorem, the algebraical characterization of (G-O) strong stability of a G-O Kuhn-Tucker point x is important. It was given by Riickmann [48] for OSI optimization and extended in [59] for our GSI one. Here, we assume EMFCQ at x. That sophisticated characterization refers to (restricted) Hessians of Lagrange functions, and it bases on a case study referring to the reduction ansatz. This RA demands strong stability in the sense of F optimization [30] for the local minimizers of the problem from the lower (y-) stage. Herewith, RA admits local representation of Vgsi(f, h, g, u, v) around x by Implicit Function Theorem [48, 59]; see [14, 61]. These cases are:
I ELICQ and RA are fulfilled at x.
II EMFCQ - but not ELICQ - and GRA are fulfilled at x.
III EMFCQ - but not GRA - is fulfilled at x.
In any case, we can also classify the type of the strongly stable stationary point x: While in case I a saddle point, a local minimizer or local maximizer is detected by the "stationary index" of x (a topological invariant), in cases II, III we have a strict local minimizer throughout [59]; cf. [31, 48, 53].
Sufficiency Part. Let CgSI 1)2)3 be satisfied. We equivalently represent PgSI(f, h,g,u,v) by pgSi(f,h,
g°,u°,v°), and straightforwardly interpret CgSI 1)2)3 as OSI conditions COsI 1)2)3. These conditions are the (OSI) constraint qualification EMFCQ, strong stability of all Kuhn-Tucker points in the sense of OSI optimization, and separateness of the values of these OSI stationary points. Under slight perturbations of the GSI data, u° ,v° do not (and need not) vary. Now, we are prepared for OSI explanations and, finally, F constructions from [23, 27, 53] in our GSI context. We briefly repeat main ideas of construction. In [27, 53], detailed information on the techniques can be found together with illustrations.
An easy counterexample shows that the separateness condition CgSI3 is not generally avoidable for establishing structural stability (see [20, 53]). Here, two connected sets, say: (arcwise) components, would have to be mapped onto one connected component, contradicting homeomorphy. A similar reasoning made for another counterexample shows that, in general, the t- (level-) dependence of the intended homeomorphisms also cannot be avoided. Moreover, each G-O Kuhn-Tucker point Xu has to be mapped to the corresponding stationary point Xd of the slightly perturbed problem VgSI(f , h, g,u, v). Finally, we conclude from the overall boundedness assumption, from EMFCQ and strong stability, that the number of G-O Kuhn-Tucker points is finite: (a G {1,... , a°}) [27, 53, 59].
We start the construction by transforming the C2-manifold M[h] to the C2-manifold M[h] in a suitable bounded, open neighbourhood of MgSI[h,g]. Therefore, first we make a local construction by a graph (or implicit function) argumentation. Locally around the stationary points, the transformation is C2. Then, we complete the whole transformation by means of a global construction. Here, we use the Morse theoretical technique of walking along trajectories of a vector field in IRn+1. Outside of (local) neighbourhoods of the stationary points, the transformation is C1. There, this means a (by 1) diminished order of differentiability, which does not cause any difficulty. From now on, we may assume that there are no inequalities, i.e., I = 0. Next, we dynamically construct the level shift —. In fact, we integrate a C^-vector field such that each critical value f (Xf) gets shifted in IR to the corresponding critical value f (Xf) (a G {1,... , a°}). Now, we may think — = IdR, referring to f o — otherwise. There are disjoint open neighbourhoods B(XU, e) (balls) around XU such that the smaller neighbourhoods B(XU, contain Xf (a G {1,... , a°}). We assume that the unperturbed and the perturbed
lower level sets coincide in all the sets B(XU, e) \ B(XU, ^) (a G {1,... , a°}). This assumption will not restrict generality.
Based on the foregoing reductions of I, — and the previous assumption, we go on constructing (t G IRn) in a local-global way. Firstly, we realize which undisturbed sets have to be homeomorphically mapped onto which corresponding sets from the disturbed situation (mapping task). We distinguish three situations given by levels t < t, t = t, or t > t. Herewith, we learn that some area from outside of the feasible set possibly has to be "carried in". Moreover, outside of the stationary points, the intersections of the level sets with the boundaries are transversal. Our further construction will be raised on these intersections (fundamental domains).
Outside of B(Xf, e) (a G {1,... , a°}), we use EMF-technique indicated in the sufficiency part on Stability Theorem. Here, we use our Lemma from Section 1, and apply this dynamical
EMF-technique on lOsi(f,g°) (= LTgSi(f,g,u,v)) and on lOsi(/ ,g°) (see Figure 4(II)). By
differential geometry, this global construction is glued together in Uf=1(B(XU, e) \ B(XU,
with the local construction sketched next. We may refer to one unperturbed stationary point xu(= xua) E {x'U,... , xUo} and corresponding perturbed point xd. Now, we are inside of B(xu,e). We restrict to n E {2, 3}, because higher dimensions can be reduced to those small dimensions by successive hyperplane intersection.
Case 1. xu is lying in the interior MOSl[g0](= Mgsi[g,u,v]) : Then, xd, being sufficiently slightly perturbed, lies in the interior of MOSl[g0]. Both stationary points are nondegenerate [19], and for each t we transform the t-levels around xu onto the local t-levels at xd. In fact, this local construction can be made by a C 1-diffeomorphism using Morse theory [27, 53].
Case 2. xu is placed on the boundary of MOSl[g0] : Then, xd may lie on the boundary or in the interior of MOSl[g0]. Without loss of generality we assume the second (boundary) case. Actually, using an implantation of a suitable level structure we turn from stationary points at the boundary to fictive stationary points in the interior. This level structure is locally given by fictive objective functions fu and fd. (In case 1, those fictive points naturally exist.) For performing this implantation of fu, fd we need precise knowledge of the configurations around the boundary points xu,xd. These configurations are characterizable by the position (relative to the boundary) of cones or balls, together with the growth behaviours of f, fg there. We have two conical types and one radial type, governed by strong stability (under EMFCQ; [27, 53, 59]. See, e.g., Figure 4(I). We arrive back in case 1 (interior position) by means of fictive interior problems, extrapolating the "characteristic" of xu, xd and implanting fictive stationary points xvjuic, xdic with their local level structures. Herewith, for all t E R the mapping task is fulfilled in case 2, too.
The delicate dynamical and topological techniques (and substeps) exhibited in Fig. 4(I) are due to the local construction in case 2. They can be elaborated, e.g., in terms of boundary displacement, positioning, sharpening or tapering flows [27, 53].
Necessity Part: Let Pgsi(f,h,g,u,v) be structurally stable. Our proof of Cgsi 1,2,3 is indirect. Assuming one of the first two regularity conditions or the third technical condition to be violated always contradicts structural stability (see Figure 5). Based on our assumptions, we carry over the proof the OSI necessity part from [23] into our GSI setting. Many details of argumentations are Morse theoretical [11, 26, 27, 53, 59]. To avoid loss of differentiability, we assume that all data are C™ [11]. This smoothness can be achieved by fine perturbations
of all OSI data and, by tracing them back, of all GSI ones.
1 2
Here, we make the inequalities of different indices Ze = Ze independent from each other (by small shifts).
Cgsi 1. As Mgsi[h,g] is compact, there exists the finite number tmax := max{f (x)| x E Mgsi[h,g]}. Herewith, Mgsi[h,g] = LTgsi(f,h,g,u,v) (t E [Tmax, to)). Moreover, we can choose perturbations slight enough such that Mgsi[h,g] remains compact. Let gmax for each sufficiently slight perturbation (f,h,g,u,v) denote the maximal (feasible) value of f. Taking t* := max{Tmax, I-1 (gmax)}, the homeomorphism <p,T* gives topological equivalence between Mgsi[h,g,u,v] = Lg*si(f,h,g,u,v) and Mgsi[h,g,u,v] = \f,h,g,u,v). By Stability Theorem, topological stability implies EMFCQ. In fact, by suitable perturbations any violation of EMFCQ at a feasible point leads to compact sets Mgsi[h, g], Mgsi[h, g], satisfying ELICQ but being not of the same homotopy type [12, 26, 53, 59]. When, e.g., the two sets have a different finite number of connected components, this must contradict topological equivalence
Cgsi 2. Suppose EMFCQ, but Cgsi2 not fulfilled: some G-O point xu be not (G-O) strongly
stable. I, a
reduction
I
b \ i i /
raising
I
mapping task fulfilled
I
Fig. 4. Proof of sufficiency part, Characterization Theorem
Perturbation Lemma [59]. Let a G-O Kuhn-Tucker point xu of VgSj(f,h,g,u,v) be given, where EMFCQ is fulfilled,, but (G-O) strong stability violated. Then, for each open C2-
neighbourhood O' of (f,h,g,u,v) there are (f ,h,g,U,V), ( f h,g,U,v) £ O' and a k' £ IN such that:
(i) Pgsx(f,h,g,U,V) has k' G-O Kuhn-Tucker points, all being (G-O) strongly stable, except one (namely, X).
(ii) VgSI( f h, g, U, V) has at least k' + 1 G-O Kuhn-Tucker points, all being (G-O) strongly stable.
(iii) In both VgSI(f ,h,g,U,V) and VgSI( f h,g,U,v), EMFCQ is satisfied everywhere, and different G-O Kuhn-Tucker points have different critical (f -or f-) values.
In F or OSI necessity parts of [11, 53] (cf. also [23]), these perturbations are realized by three steps. Step 1 yields local isolatedness of x as a stationary point where, additionally, (E)LICQ is guaranteed but unstability preserved. In step 2, outside of the local situation, (E)MFCQ and strong stability of all (other) stationary points are established. Finally, in step 3, the unstable Kuhn-Tucker point Xu "splits": By this bi- (or tri-) furcation we locally get two new stationary points; they have strongly stability. In this GSI situation, we use the algebraical characterization from our preparations. Now, we introduce a topological idea: For LrgSX(f ,h,g,U,V), LrgSX( f,h, g,U,v) we have to take into account each change of the homeomorphy type of a lower level set, when t traverses (-to, to). Based on the perturbations
from above, we apply the following items on VgSI(f ,h,g,U,V), and VgSI( f, h, g, U, V). We
look at a C2-problem (f ,h,g,u,v) having a compact feasible set and fulfilling EMFCQ, and we put LbgSI«(/ ,h,g,u,v) := {x £ MgSI[h, g]| a < / (x) < b} for some a,b £ R,, a < b.
Item 1. If a(f ,h,g,u,v) does not contain a stationary point, then LgSI(f ,h,g, u,v) and LgSI(f ,h,g,u,v) are homeomorphic.
Item 2. Let LgSIa(f,h,g,u,v) contain exactly one stationary point X'. Moreover, let a < f (X') < b and X' be (G-O) strongly stable. Then, LgSI(f ,h,g,u,v) and LbgSI(f ,h,g,u,v) are not homeomorphic.
These two items immediately result from corresponding facts on (f,h,g0,u°,v0),
( £ h, ~g°, u°, v0) stated in [48]. Here, Item 2 can be expressed with attaching «-cells (k = stationary index at X'; [59]). By Manifold Theorem and Lemma (Sections 1-2) we conclude for all noncritical levels t: LgSI( f, h,g,u,v) = MgSI[h, (g, — f + t)] is a compact topological manifold (with boundary). So, their homology spaces (over R) are of different finite dimensions [51]. As these spaces are topological invariants, the two considered lower level sets cannot be homeomorphic [19].
Now, we can make the following "discrete" statement on numbers of topological changes for the lower level sets: The homeomorphy type of L£SI( / h, g, u, v) changes (at least) at k' + 1 times, while the homeomorphy type of LgSI( f,h,g,u,v) changes (at least) at kk — 1 times, but at most at kk times. This difference contradicts structural stability of (f,h,g,u,v) (cf. [59], or see Fig. 5).
3: Let CgSI 3 be violated, but the former two properties on EMFCQ and strong stability be satisfied. By local addition of arbitrarily small constant functions on f, we get a problem (f *,h, g,u,v) satisfying CgSI 3. Let k* stand for the number of critical points of (f*,h,g,u,v). Then the homeomorphy type of LgSI(f*,h,g,u,v) changes k* times, while the number of changes of the homeomorphy type of LgSI(f,h,g,u,v) is less than k*. Hence, we are faced again with a situation which is incompatible with structural stability of PGsi(f,h,g,u,v) (Figure 5).
Fig. 5. Proof of necessity part, Characterization Theorem
4. Generalizations, Optimal Control and Conclusion. 4.1. Generalizations.
There are two lines for generalizing our topological results: (i) Mgsi[h,g] is unbounded (noncompactness),
(ii) f is of the nondifferentiable GSI maximum-type f (x) = max7eY(x) w(x,Y).
On (i): We overcome noncompactness by turning to the entity of excised subsets of MgSI[h, g]. Roughly speaking, the effect of intersection is performed by subtracting lower semi-continuous functions from hi (i £ I) and g [49, 53, 59]. Herewith, we can express cuts, e.g., by cylinders or balls, by IRn itself or by bizarre sets. Referring to all excised sets, we get the condition of excisional topological stability which can actually be characterized by the overall validity of EMFCQ in the unbounded set MgSI[h,g]. For that (Excisional) Stability Theorem see [59].
On (ii): Nonsmoothness is overcome by expressing PgSI(f,h,g,U,V) as minimization of xn+i over the epigraph EgSi(f) := {(x,xn+1)| x £ MgSi[h,g], f (x) < xn+1}. From this problem in IRn+1 we obtain our stationary points of PgSI(f,h,g,U,V) and the appropriate condition of strong stability [53, 54, 59]. Now, (max-) structural stability of our nondifferentiable problem can be characterized by EMFCQ, strong stability and the technical separateness condition again. This Characterization Theorem and the one for the case combination of (i) and (ii) are demonstrated in [59].
4.2. Optimal Control of Ordinary Differential Equations.
We turn to infinite dimensions by studying the following minimization problem in (x,u) [13, 36, 44]:
P(£, L,F,H, G) <
Min I(x,u) := £(x(a),x(b)) + j L(t,x(t),u(t)) dt (x £ (Cpw 2 ([a, b],R))n, u £ (FpW 2 ([a,b],R))q ),
such that
x(t) = F(t,x(t),u(t)) (for almost every t £ [a,b]), (x(a),x(b)) £ M[H], K x(t) £ MF[G(t, ^u(t))] (for almost every t £ [a,b]),
where (L,F,G), (£,H) are C3- and C2-functions (vector notation), respectively. Instead of referring to the larger classes of Sobolev or Lebesgue spaces, we concentrate on spaces of continuous and piecewise C2 states x, and piecewise C2 controls, called C0W2 and FpW2. For these spaces, strong topologies in Whitney's sense can be generally introduced [59].
Assumption (BOUND). M[H] C Rn x Rn and MF[G] C [a,b] x Rn x Rq, defined by the equality and inequality contraints, are bounded.
Assumption (LB). There exist positive functions £ C(IRq+1, R) such that (under
|| • = maximum norm) we have linear boundedness of F:
||F(t, x, u)|U < a0(t, u)||x|U + Po(t, u) ((t, x, u) £ Rn+q+1).
We briefly present two approaches to global structure and stability of P (£,L, F, H, G) (cf. [59], where a third one can also be found). While our main Approach II is refined, Approach I is given for a better understanding.
Approach I: Particular Structure. Let u be considered as C2 and a parameter. Then, for each fixed u = u* the optimal control problem P(£, L, F, H, G) becomes a problem Pu* (£, L, F, H, G) from calculus of variations. The corresponding system of differential equations (on x) generates a flow (in IRn+1; [1, 20]). Under this flow, we trace back the equality and inequality constraints, and the objective functional as well (cf. [55, 56, 59]). So we obtain an
OSI problem V'Usi(f*,h*,g*) (where Yj = [a,b]). Then, referring to the family of all u and to perturbations of (f*,h*,g*), we get the condition of (particular) structural stability with its Characterization Theorem again (cf. Section 3; [56, 59]). The C2-property and parametrical treatment of u, however, are not sufficient for optimal control. That is why we turn to the
Approach II: Composite Structure. We evaluate the necessary optimality condition Pontryagin's minimum principle [13, 44] in the way of "Kuhn-Tucker" for almost every t £ [a,b]. Here, we have suitable multiplier vectors, (adjoint) variables, and H(t, x, u,A,^) := L(t, x, u) — ATF(t, x, u) — ^TG(t, x, u). Then, our evaluation, called minimum principle here [6, 38, 39, 41], reads
DUH(t,x0(t),u0 (t),A0(t),^0(t)) = 0q, jt) > 0 (j £ J) and (t) G(t,x0(t),u0(t)) = 0, A0(a) = — D^ (£ — poTH)(x0(a),x0(b)), A0(b) = DX2 (£ — poT H )(x0(a),x0(b)), A0(t) = —DXH(t, x0(t),u0(t), A0(t), (t)).
For our causal (composite) structure we need a condition like strong stability [59]:
Assumption (CONT). All the (C0w2 x Fpw2-) solution components (x0,u0) of the minimum principle depend continuously on Cf x Cf-perturbations ((L,F,G), (£,H)) ^
((L,F,G), (£,H)).
We interpret the first four lines of the minimum principle as Kuhn—Tucker conditions of two families of optimization problems: (*) (L,F — w,G) and (**) V?(A0(a), A0(b),£, H),
an index set Mp° [F,G] of (t, x, w) being appropriately chosen in view of V(£, L, F, H,G). For each of these problems we introduce (composite) structural stability and characterize it essentially by (E)MFCQ and strong stability (see Section 3). Analyzing (*) so, we locally get implicit C2-control functions u^(t, x, w), which are Kuhn-Tucker point-valued and fulfill u0(t) = u^(t,x0(t),x(t)). Substituting w := x(t) for any trajectory x of some auxiliary flow, adapted to our system of differential equations, we locally receive core functions ~Uy(t, x). The choices of these auxiliary or test flows etablish a structural frontier of our theory [59]. In order to globalize a core such that its domain covers [a,b], we admit jumps in Rn+1 (see Figure 6). These jumps shall be compatible with the jumps of our variables u0. Again we say that the globalized core functions (ty) u° are of class Fpw 2. Let B, B be ("boundary") sets where the jumps may or really do happen, respectively. When these sets exist as Lipschitzian manifolds of dimension q, and if they define (by decomposition) piecewise structures before or after jumps, which quantitatively remain preserved under small perturbation of (£, L, F, H, G), then the core (ty) is called (composite) structurally stable [59]. A further regularity condition, called structural transversality, in short: ST, analytically determines the boundary sets (up to a finite number of choices) and guarantees this (composite) structural stability of a core. (See also [21, 39, 41].) The refined condition ST essentially means transversal intersection of uy(^,x(^)) (along trajectories x) at the boundary of the corresponding feasible set in Rq. This implies transversality of x at the manifolds B, B.
Now, inserting u(t) = u^j(t,x(t)) in V(£, L, F, H,G) delivers again a problem VUy (£,L, F, H, G) from calculus of variations, which we also trace back under its flow. In this way we get an optimization problem with a complex underlying piecewise structure. Up to the structural frontiers given by combinatorially more complicate index sets Y(x) and objective functions f of continuous selection type [22], we arrive at a GSI problem (***) VgSI(f, h,g,v) with f of
u,
u
20.
0
Uv :
Fig. 6. Piecewise structure and jumps of cores
maximum-type (cf. Subsection 4.1). Then, we introduce this optimization problem's condition of (composite) structural stability referring to perturbations of the original data (£, L, F, H, G).
In that sense, we call P(£, L, F, H, G) composite structurally stable if all the structural elements (*), (**), (***), (V) are (composite) structurally stable. Under our basic Assumptions (BOUND), (LB) and up to those more complex problems we state (with simplified presentation):
Characterization Theorem on Composite Structural Stability [59]. The problem P(£, L, F, H, G) is composite structurally stable, if and only if the conditions Ci, 2,3,4 are satisfied:
Ci. (E)MFCQ holds for all the feasible sets underlying (*), (**), (* * *), (V).
C2. All the Kuhn-Tucker points u, x of the problems represented in (*), (* *), (* * *) are strongly stable (in F or G-O sense).
C3. For all optimization problems represented in (*), (**), (* * *) each two different Kuhn-Tucker points have different (separate) critical values.
C4. For all core functions (V) ST is fulfilled.
Sketch of Proof: The main lines are the same as in Subsection 3.3. The new item, given in the necessity part, " C4," concerns the undisturbed or disturbed piecewise structures, and it is illustrated in Figure 7.
For controllability, i.e., to come from time a to time b under given constraints of P(£, L, F, H, G), discrete mathematics [5] often turns out to be a tool of investigation as follows. (For underlying finiteness and genericity considerations see [59].) Our control problem asks for a domain of the core (compatible with u0) that is sufficiently large, say: tending to maximality. Provided a careful choice of the set of jumps, this maximal domain problem can be represented as a maximal matching problem in a partite graph (see, e.g., Figure 8). In a
subset of arcs called matching, different elements are disjoint. Here, each partition stands for a locally defined continuous core, the directedness of the ars reflects orientation by time t. This matching problem can be solved by Edmond's algorithm [25].
Inserting the global cores, arriving at an x-depending problem, we may, for example, consider
the objective function as the arc length of our piecewise structured solutions x = x° G CP
pw 2
of minimum principle. Therefore, we take into account arcs between neighbouring vertices (manifolds Bi, B2) of the same former partition such that the partite character gets lost (see, e.g, Figure 9, periodic constraint x0(a) — x0(b) = 0 implied). The corresponding minimization problem can be regarded as a shortest path problem, solvable by Dijkstra's algorithm [25].
Fig. 7. Proof of necessity part (composite structural stability)
Fig. 8. Tripartite directed graph featuring controllability problem
Fig. 9. Directed graph featuring minimization of arc length
4.3. Conclusion.
Besides the " discrete" (stationary) index, piecewise structures and optimization problems mentioned above, there is a variety of further theoretical and methodical connections between GSI optimization, optimal control and discrete mathematics. Concerning discrete Morse theory, topology and systems analysis we mention [2, 8] and [42]; many other examples can be found in [35, 59, 60]. We noted that structural frontiers can often be understood in a combinatorial manner. Just the same is true with respect to solution algorithms for a given GSI problem. The more complex the problem is, the more important becomes discrete intrinsic information of the problem for transparency, convergence and stability (cf. [52], see also [34] on timeminimal control). Here, we refer to the research [43, 58, 59] that is based on our optimality conditions, analytical techniques and stability results. Often, there is little knowledge about the geometry and topology of the feasible GSI set. As a "manifold" and stability condition, our version of EMFCQ, that bases on Assumption Byo of LICQ, was of central importance for a rigorous study of optimization and control. In future, there may be both a weakening in the theoretical field of assumptions (e.g., in the way of [24]) and a systematic look for combinatorial and geometrical treatments from reverse engineering, discrete tomography with its inverse problems, or randomization [59].
Acknowledgement. The author thanks Prof. Dr. W. Krabs and Prof. Dr. Yu. Shokin for encouragement, Prof. Dr. B. Lemaire for encouraging submission of a related survey article, and Dipl.-Math. S. Mock for technical help.
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Received for publicatuion December 22, 2000
